Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 159

Ǜ P C Ǜ g i I =x 0
Ǜ P C Ǜ g C i I =x 0

. M C I /
. M I /

exp .2 I / D ; (5.45)

where

ik ? L
x 0

C k ? tanh .k ? d/
k ? C tanh .k ? d/ :

Ǜ g D (5.46)

The functions M , I , and Ǜ g given by Eqs. ( 5.17 ), ( 5.20 ), ( 5.30 ), and ( 5.46 )
depend on k ? . This means that Eq. ( 5.45 ) defines an inexplicit dependence of
dimensionless frequency on perpendicular wave number, i.e., x 0 D x 0 .k ? /.
The range of very small values of k ? ,ork ? L 1, is not of great importance for
practical applications since this limit corresponds to the large-scaled perturbations
k 1

? L 10 3 km. Furthermore, these scale sizes are of the order of the
Earth radius and thus cannot be considered in the framework of the plane-stratified
model developed in this section. So, we restrict our analysis to the opposite case of
k ? L 1. This implies that the typical size of FMS waves propagating inside the
IAR is much smaller than the resonator scale L. Not surprisingly, these waves have
a dispersion relation as though they were in an infinite space. In other words, one
may expect that the dispersion relation of the FMS wave has to be insensitive to the
boundary condition at the resonator walls. Indeed, noticing that formally the value
I D 0 satisfies Eq. ( 5.45 ) because in this case both parts of Eq. ( 5.45 ) are equal to
unity, we come to the equation

x 0 D k ? L or ! D k ? V AI ; (5.47)

which corresponds to the ordinary dispersion relation ( 2.26 ) for the FMS mode
in a homogeneous plasma. Nevertheless, this relation can be considered only as a
first approximation because it does not satisfy the original equation ( 5.44 ). One can
expand Eq. ( 5.45 ) in a power series of small parameter .k ? L/ 1 to find the function
! D !.k ? / in the next approximation (Surkov et al. 2004 ).

5.2.3 Mode Coupling: The Role of the Ionospheric

Hall Conductivity

In the above consideration the shear and FMS modes are uncoupled. It is a
conventionally idealized model which can be relevant for the nighttime conditions.
However, during daytime conditions, the situation may become more complex since
the effects due to finite Hall conductivity start to play an important role. Analysis
of the general dispersion relation equation ( 5.36 ) has shown that despite the strong
mode coupling equation ( 5.36 ) consists of two branches as before (Surkov et al.
2004 ). Each branch includes a discrete set of normal modes. In Fig. 5.5 we plot